To find side $c$ in a triangle where $\angle B = 28^\circ$, $\angle C = 52^\circ$, and side $b = 45$, we can use the Law of Sines, which is given by:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Step 1: Find $\angle A$

Since the sum of angles in a triangle is always $180^\circ$, we can find $\angle A$ by subtracting the sum of $\angle B$ and $\angle C$ from $180^\circ$:

$\angle A = 180^\circ - (28^\circ + 52^\circ) = 100^\circ$

Step 2: Apply the Law of Sines

We know:

• $b = 45$
• $\angle B = 28^\circ$
• $\angle C = 52^\circ$

Using the Law of Sines, we get:

$\frac{c}{\sin C} = \frac{b}{\sin B}$

Now substitute the known values:

$\frac{c}{\sin 52^\circ} = \frac{45}{\sin 28^\circ}$

Step 3: Solve for $c$

First, calculate $\sin 52^\circ$ and $\sin 28^\circ$:

$\sin 52^\circ \approx 0.7880, \quad \sin 28^\circ \approx 0.4695$

Now, plug these into the equation:

$\frac{c}{0.7880} = \frac{45}{0.4695}$

Simplify:

$\frac{c}{0.7880} = 95.87$

Finally, solve for $c$:

$c = 0.7880 \times 95.87 \approx 75.52$

Conclusion:

Side $c \approx 75.52$.

Would you like further details or have any questions?

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Here are 5 related questions:

1. How do we apply the Law of Cosines in such problems?
2. What if we are given the sides instead of angles?
3. How do you solve for an unknown angle using the Law of Sines?
4. What happens if the triangle is non-obtuse or obtuse?
5. What if one of the angles is $90^\circ$?

Tip: Always ensure the triangle satisfies the sum of angles rule when solving any triangle problem.